On Sunday, 16 June 2013 11:25:27 UTC+2, fom wrote:> But, you respect neither mathematics based upon axioms nor logic based upon

contradictions. Like the undefinability of elements and extensionality:

"Eventually, most mathematicians came to accept that definability should not be required, partly because the axiom of choice leads to nice results, but mostly because of the difficulties that arise when one tries to make notion of definability precise." (Andreas Blass)

That is a real surprise to me. Which mathematicians accepted that and when? Was there a public meeting with voting like in meta or like in the astronomy scene when Pluto has been degraded?

Wouldn't a set with undefined elements contradict the Axiom of Extensionality:

If every element of X is an element of Y and every element of Y is an element of X, then X = Y.

How could that be decided for undefined elements?

But my actual question is this: I have heard (but don't remember where) that there is another solution: The set of finite definitions is countable. That cannot be explained away, can it? But not every finite definition has a meaning. In fact, if we refrain from using common sense, we cannot even define definability, let alone the set of meaningful definitions. Therefore this set is not countable but subcountable - and if we identify subcountability with uncountability, we have won and can continue to enjoy the nice results of the axiom of choice.

Obviously to vague formulated as that matheologians could understand it - with their precisely defined definitions. Therefore deleted in MO after an hour.